Robust Input Verification for Secure Multi-Party Computation (MPC) with Clients

ABSTRACT

In one set of embodiments, each server executing a secure multi-party computation (MPC) protocol can receive shares of inputs to the MPC protocol from a plurality of clients, where each input is private to each client and where each share is generated from its corresponding input using a threshold secret sharing scheme. Each server can then verify whether the shares of the plurality of inputs are valid/invalid and, for each invalid share, determine whether a client that submitted the invalid share or a server that holds the invalid share is corrupted. If the client that submitted the invalid share is corrupted, each server can ignore the input of that corrupted client during a computation phase of the MPC protocol. Alternatively, if the server that holds the invalid share is corrupted, each server can prevent that corrupted server from participating in the computation phase.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent application Ser. No. 17/010,526, filed Sep. 2, 2020 and entitled “ROBUST INPUT VERIFICATION FOR SECURE MULTI-PARTY COMPUTATION (MPC) WITH CLIENTS,” the entire contents of which are incorporated herein by reference for all purposes.

BACKGROUND

Secure multi-party computation (MPC) is a cryptographic technology that enables two or more parties to jointly compute a function over a set of inputs while keeping the inputs private. For example, consider a simple scenario in which three parties A, B, and C have respective private inputs x, y, and z and would like to obtain the sum of these inputs without revealing x, y, and z to each other. In an ideal world, A, B, and C could each submit their input to an incorruptible and perfectly trustworthy third party

.

could then compute x+y+z on their behalf and output the resulting sum, thereby preventing A, B, and C from learning each other's inputs. In the real world, however, incorruptible and perfectly trustworthy third party

does not exist. MPC addresses this problem by providing protocols which allow A, B, and C to collectively evaluate, via a series of message exchanges, x+y+z in a manner that achieves the same input privacy as the ideal world model, without relying on

.

A common approach for designing an MPC protocol involves having each party distribute cryptographic shares of its input to the other parties using a threshold secret sharing scheme such as Shamir's secret sharing. Through this process, each party receives partial information regarding the other parties' inputs, which is sufficient for the party to carry out the protocol's computation but is insufficient (at least in isolation) for the party to learn the inputs. In an MPC protocol executed by n parties, the sharing of an input x via a secret sharing scheme with threshold t is secure in the sense that up to t of the n parties may collude and disclose their shares to each other without learning anything regarding x. Such colluding parties are referred to as “corrupted” and it is assumed that all corrupted parties follow a single strategy dictated by a “semi-honest” or “malicious” adversary. If the adversary is semi-honest, the corrupted parties will correctly follow the MPC protocol specification and will only try to learn private information by collecting data arising out of normal protocol execution (e.g., the internal state of each corrupted party, the transcript of messages received, etc.). If the adversary is malicious, the corrupted parties may arbitrarily deviate from the protocol specification in order to learn private information. MPC protocols that can withstand attacks from semi-honest adversaries or malicious adversaries are said to “be semi-honest-secure”/“have semi-honest security” or “be malicious-secure”/“have malicious security” respectively.

In recent years, a number of secret sharing-based MPC protocols with malicious security have been proposed that are close in computational and/or communication cost to semi-honest-secure MPC protocols. However, these proposed protocols suffer from at least two deficiencies. First, they are “secure with abort” (or in other words, are “non-robust”), which means that the adversary can cause the protocol to abort prior to completion, thereby preventing uncorrupted (i.e., honest) parties from obtaining the protocol's output. Second, the proposed protocols only allow parties that participate in protocol computation (referred to herein as “servers”) to submit inputs. This limitation—which arises out of the protocols' inability to distinguish between corrupted clients and corrupted servers at the time of verifying input sharings—is problematic because many MPC applications (e.g., secure auctions, electronic voting, etc.) are designed to receive inputs from a large group of “clients” that do not take part in computation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts an example operating environment in which embodiments of the present disclosure may be implemented.

FIG. 2 depicts a flowchart for implementing non-batched robust input verification according to certain embodiments.

FIGS. 3A and 3B depict a flowchart for implementing batched robust input verification according to certain embodiments.

DETAILED DESCRIPTION

In the following description, for purposes of explanation, numerous examples and details are set forth in order to provide an understanding of various embodiments. It will be evident, however, to one skilled in the art that certain embodiments can be practiced without some of these details or can be practiced with modifications or equivalents thereof.

1. Overview

The present disclosure is directed to techniques for implementing robust input verification in a secret sharing-based malicious-secure MPC protocol that is capable of receiving inputs from N clients (and is executed by n servers). As used herein, “input verification” refers to the task of verifying that the shares of an input provided by a client to the servers via a threshold secret sharing scheme are valid, which means that the shares conform to the secret sharing scheme's rules/restrictions.

The techniques of the present disclosure rely on two assumptions: first, at most t<n/4 servers are corrupted (and thus at least 3t+1 servers are honest), and second, at most (1−ρ)N clients are corrupted for 0<ρ≤1 (and thus at least ρN clients are honest). With these assumptions in place, the techniques ensure that (1) the adversary controlling the corrupted clients and corrupted servers cannot abort execution of the protocol, (2) a corrupted client cannot disqualify an honest server from participating in the protocol, and (3) a corrupted server cannot “censor” an honest client (i.e., prevent that client's input from being included in the protocol's computation). The foregoing and other aspects are explained in further detail in the sections that follow.

2. Operating Environment and High-Level Solution Description

To provide context for the embodiments described herein, FIG. 1 depicts an operating environment 100 comprising N clients C¹, . . . , C^(N) (reference numerals 102(1)-(N)) and n servers S₁, . . . , S_(n) (reference numerals 104(1)-(n)) and a general framework that may be used by the clients and servers for executing a secret sharing-based malicious-secure MPC protocol. As shown, this framework comprises three main phases: an input phase 106, an input verification phase 108, and a computation phase 110.

During input phase 106, each client C^(k) submits an input x^(k) to servers S₁, . . . , S_(n) by sharing it using a threshold secret sharing scheme, which means client C^(k) submits a cryptographic share x_(q) ^(k) of x^(k) to each server S_(q) for q=1, . . . , n. If Shamir's secret sharing is used, input x^(k) is an element of a finite field

and each share x_(q) ^(k)=f(q) where f is t-degree polynomial with f(0)=x^(k) and where t is the maximum number of corrupted servers. It is well known that such a t-degree polynomial (or “t-polynomial”) theoretically hides the secret encoded at f(0) from any subset of at most t shareholders. Vector {right arrow over (x^(k))}=(x₁ ^(k), . . . , x_(n) ^(k)), which represents the shares of input x^(k) provided to servers S₁, . . . , S_(n), is said to be “consistent with,” or a “consistent sharing” of, x^(k) if the shares of all honest servers agree with polynomial f (i.e., reside on the polynomial, such that x_(q) ^(k) =f(q) for every honest server S_(q)). By a corollary, there are at most t servers that disagree with the polynomial in this scenario. Further, vector {right arrow over (x^(k))} is said to be a “perfectly consistent with,” or a “perfectly consistent sharing” of, x^(k) if the shares of all servers (even corrupted ones) agree with polynomial f. For the remainder of this disclosure, [x] is used to denote a consistent sharing of a given input/secret x.

Once all clients have provided their input shares, servers S₁, . . . , S_(n) verify (via input verification phase 108) that the shares of each input are valid in order to detect cheating by corrupted clients/servers. The servers perform this verification in a manner that does not reveal the shares to each other, thereby maintaining the privacy of inputs x¹, . . . , x^(N). There are several ways in which input verification phase 108 can be generally structured based on, e.g., the specific threshold secret sharing scheme used, degree of complexity tolerated, and so on. If Shamir's secret sharing is employed as mentioned above, one approach is for each server S_(q) to obtain a share r_(q) of a secret random value r where the sharing of r is guaranteed to be consistent. For example, shares r₁, . . . , r_(n) can be computed offline and seeded on respective servers S₁, . . . , S_(n) using a secure method. Each server S_(q) can then (1) add r_(q) to its share x_(q) ^(k) of a client input x^(k) to generate a_(q) ^(k), (2) broadcast a_(q) ^(k) to every other server, and (3) upon receiving the other servers' a values, invoke a Reconstruct([a]) procedure which attempts to interpolate a t-polynomial f such that f(i)=a_(i) ^(k) for i=1, . . . n.

If the Reconstruct procedure is successful, that means {right arrow over (x^(k))} is a consistent sharing of input x^(k) and the shares which agree with interpolated polynomial f are valid shares of x^(k). Conversely, the shares which do not agree with f in this scenario are invalid shares of x^(k) and are referred to as “bad points” (because they do not reside on the polynomial). In general, the Reconstruct procedure will succeed in interpolating f if there are at most

$\left\lfloor \frac{n - t}{2} \right\rfloor$

disagreeing shares (or stated another way, if there are at most

$\left\lfloor \frac{n - t}{2} \right\rfloor$

shares which do not reside on f).

Finally, during computation phase 110, servers S₁, . . . , S_(n) compute the function of the MPC protocol over the input shares verified during input verification phase 108 and obtain a protocol output. This specific nature of this computation, which is beyond the scope of this disclosure, will vary based on the characteristics of the function and the protocol implementation.

As noted the Background section, there are a number of secret sharing-based malicious-secure MPC protocols in the art that offer relatively high efficiency, but are non-robust (i.e., can be aborted by the adversary) and are incapable of receiving inputs from non-participant clients. A key reason for the latter limitation is that, when a client input is shared among servers and cheating (i.e., an invalid share) is detected during input verification phase 108, these existing protocols cannot easily determine whether the cheating was carried out by a corrupted client or a corrupted server, which raises significant problems. For example, if the cheating was performed by a corrupted client but it is erroneously blamed on an honest server, that server will be excluded from further participation in the protocol, which means that corrupted clients can arbitrarily disqualify honest servers. If such an attack is repeated n−t times then only corrupted servers will remain for carrying out computation phase 110, which is clearly no longer secure (as corrupted servers may collude and reveal their shares to each other). Conversely, if the cheating was performed by a corrupted server but it is erroneously blamed on an honest client, that client's input will be ignored during the computation phase, which means that corrupted servers can arbitrarily censor honest clients.

To address the foregoing limitations, embodiments of the present disclosure provide a novel input verification protocol—usable by servers S₁, . . . , S_(n) during input verification phase 108 of FIG. 1 —that (1) is robust, (2) ensures corrupted clients cannot disqualify honest servers, and (3) ensures corrupted servers cannot censor honest clients. At a high level, properties (2) and (3) are achieved by assuming at most (1−ρ)N clients are corrupted (and thus corrupted clients can “blame” honest servers, by dealing them invalid shares, at most (1−ρ)N times). With this assumption, the novel input verification protocol can distinguish between corrupted clients and corrupted servers during input verification phase 108, thereby avoiding the incorrect disqualification of honest servers or the incorrect censoring of honest clients.

For example, as described in further detail below, the novel protocol can initialize a counter ctr_(q) to 0 for each server S_(q). The protocol can then increment this counter each time S_(q) is found to hold an invalid share/bad point for a consistent input sharing (per, e.g., Shamir's Reconstruct procedure). If ctr_(q) exceeds (1−ρ)N at any juncture of input verification phase 108, the protocol can definitively conclude that S_(q) is corrupted and can eliminate it from participating in computation phase 110. For input sharings that are found to be consistent but include some number of bad points held by servers with counters less than (1=ρ)N, the novel protocol can request that the clients which originated those sharings resubmit them via a stronger (and more expensive) method to definitively determine whether the clients or servers are corrupted.

With regard to property (1) above (i.e., robustness), this is achieved via a combination of properties (2) and (3) and the assumption that at most t<n/4 servers are corrupted. As mentioned previously, with Shamir's secret sharing, the Reconstruct procedure will succeed in interpolating a t-polynomial f for a sharing of a given input/secret x if there are at most

$\left\lfloor \frac{n - t}{2} \right\rfloor$

disagreeing shares/servers. When n≥4t+1, this procedure succeeds if there are at most 1.5t bad points (i.e., invalid shares/disagreeing servers). Accordingly, the adversary cannot abort the protocol by, e.g., causing some combination of corrupted servers/clients to broadcast/submit invalid shares (or nothing at all) for a given input x^(k) to the honest servers, because (A) if at least 3t+1 servers (i.e., the honest servers) receive valid shares, Reconstruct[a^(k)] will succeed (and the protocol can proceed to identify corrupted servers as above), and (B) if Reconstruct[a^(k)] fails due to more than n/4 bad points, client C^(k) can be definitely identified as corrupted. In either case, the protocol can move forward and thus avoid being halted.

Section (3) below presents a “non-batched” implementation of the novel input verification protocol in which verification is performed on a per-client basis (i.e., sequentially on each input sharing submitted by clients C¹, . . . , C^(N)). And section (4) below presents a more efficient “batched” implementation of the novel input verification protocol that optimistically assumes all clients and servers are honest and thus attempts to verify the input sharings of all clients C¹, . . . , C^(N) together (i.e., as an aggregation). This is possible because, under Shamir's secret sharing, if two or more sharings are combined via a linear operator and those individual sharings are perfectly consistent, the combination is also guaranteed to be perfectly consistent. If the verification of the aggregated sharing of C¹, . . . , C^(N) succeeds (i.e., the aggregated sharing is perfectly consistent), servers S₁, . . . , S_(n) can immediately move on to computation phase 110. Alternatively if the verification of the aggregated sharing fails (i.e., the aggregated sharing comprises one or more invalid shares), the servers can perform a binary search to find the invalid shares (i.e., verify aggregations of the first and second halves of the shares), and this can continue recursively until the invalid shares are identified.

3. Non-Batched Robust Input Verification

FIG. 2 depicts a flowchart 200 that may be executed by each server S_(i) of FIG. 1 (where i=1, . . . , n) for implementing a non-batched (i.e., per-client) version of the novel input verification protocol of the present disclosure according to certain embodiments. Flowchart 200 assumes that the inputs submitted by clients C¹, . . . , C^(N) are shared via Shamir's secret sharing; however, one of ordinary skill in the art will appreciate that the high-level concepts embodied by this protocol may also be applied to scenarios in which other threshold secret sharing schemes are used.

Starting with blocks 202 and 204, server S_(i) can receive shares x_(i) ¹, . . . , x_(i) ^(N) of inputs x¹, . . . , x^(N) from clients C¹, . . . , C^(N) and can initialize a counter ctr_(q) for every server S_(q) (where q∈[n]) to zero.

At block 206, server S_(i) can initialize a set of corrupted servers {tilde over (S)}, a set of corrupted clients {tilde over (C)}, a set of blocked clients Ĉ, and a set of honest clients H to null (0). It is important to note that during execution of the protocol all honest servers agree on these three sets (so for example, there is no situation of a conflict in which honest server S_(i) decides that server S_(c) is corrupted while honest server S_(j) decides that server S_(c′) (c≠c′) is corrupted). Server S_(i) can then enter a loop for each input x^(k) provided by a given client C^(k) for k=1, . . . , N (block 208).

Within the loop, server S_(i) can obtain a share r_(i) ^(k) of a sharing of a secret random value r^(k) where the sharing is guaranteed to be consistent (block 210). As mentioned previously, in certain embodiments shares r₁, . . . , r_(n) can be computed offline and seeded on respective servers S₁, . . . , S_(n) using a secure method.

Upon obtaining r_(i) ^(k) , server S_(i) can compute a_(i) ^(k)=r_(i) ^(k)+x_(i) ^(k) (block 212), securely broadcast a_(i) ^(k) to the other servers (block 214), and receive share a_(q) ^(k) from every other server S_(q), thereby allowing it to build a sharing {right arrow over (a^(k))} (block 216).

At blocks 218 and 220, server S_(i) can invoke Reconstruct([a^(k)]), which attempts to find/interpolate a t-polynomial f in which f(q)=a_(q) ^(k) for at least 3t+1 shares of {right arrow over (a^(k))}, and check the outcome. If the Reconstruct procedure fails (i.e., no polynomial is found), server S_(i) can conclude that client C^(k) is corrupted and can add client C^(k) to the set of corrupted clients {tilde over (C)} (block 222).

If the Reconstruct procedure succeeds but there is a set of servers S′⊂S whose shares disagree with reconstructed polynomial f (i.e., the sharing is consistent), server S_(i) can increment counter ctr_(q) for each server S_(q)∈S′ (block 224). As part of this step, if there is a server S_(q) whose counter ctr_(q) has exceeded (1−ρ)N, server S_(i) can add server S_(q) to the list of corrupted servers {tilde over (S)}.

And if the Reconstruct procedure succeeds and there are no bad points (i.e., the sharing is perfectly consistent), server S_(i) can add client C^(k) to the set of honest clients H (block 226). Server S_(i) can then reach the end of the current loop iteration (block 228) and return to block 208 to process the next input x^(k).

Upon processing all client inputs, server S_(i) can populate the set of blocked clients Ĉ with those clients that are not in {tilde over (C)} or H (block 230). Finally, at block 232, server S_(i) can output the set of blocked clients Ĉ, the set of corrupted clients {tilde over (C)}, and the set of corrupted servers {tilde over (S)} and flowchart 200 can end. Generally speaking, the inputs of corrupted clients (i.e., clients in {tilde over (C)}) will be disregarded by the servers during computation phase 110 and the corrupted servers (i.e., servers in {tilde over (S)}) will be blocked from participating in phase 110.

Further, although not specifically shown in FIG. 2 , the blocked clients in Ĉ can be given another opportunity to provide their respective shares via an alternative, stronger input scheme in which client-side cheating is disabled by design. This stronger input scheme will typically require more communication and/or compute resources than the input scheme described above but should not significantly impact the overall performance of input verification phase 108 if (1−ρ) is kept relatively small.

4. Batched Robust Input Verification

FIGS. 3A and 3B depict a flowchart 300 that may be executed by each server S_(i) of FIG. 1 (where i=1, . . . , n) for implementing a batched version of the novel input verification protocol of the present disclosure according to certain embodiments. As noted above, instead of verifying the sharing of each client individually, this batched approach attempts to verify the input sharings of all clients C¹, . . . , C^(N) together (i.e., as an aggregation), utilizing the fact that the addition of perfectly consistent sharings results in a perfectly consistent sharing.

This batched approach results in at least two efficiencies: first, if all clients and servers are honest (which will often be the case in many applications), only a single verification round is needed. This is in contrast to the non-batched approach presented in FIG. 2 where every client input must be verified, regardless of whether all clients/servers are honest or not. Second, the batched approach requires that the servers communicate with each other only once during input verification phase 108 (in order to communicate all of their shares for all client inputs). With the non-batched approach of FIG. 2 , the servers need to communicate with each other for every client input in order to verify that particular input.

It should be noted that the batched approach assumes that (1−ρ) is relatively low (i.e., there are relatively few corrupted clients). In cases where (1−ρ) approaches 1, it may be preferable from a performance perspective to perform per-client verification per flowchart 200 of FIG. 2 . It is possible to bound the value of (1−ρ)N using various methods such as, e.g., a proof-of-work based solution that prevents a single adversarial entity from efficiently coordinating many malicious clients, a required per-client payment, and so on. Bounding (1−ρ)N remains out of the scope of this disclosure.

Turning now to FIG. 3A, starting with blocks 302 and 304, server S_(i) can receive shares x_(i) ¹ , . . . , x_(i) ^(N) of inputs x¹, . . . , x^(N) from clients C¹, . . . , C^(N) and enter a loop for each client C^(k) (where k=1, . . . , N). Within this loop, server S_(i) can obtain a share r_(i) ^(k) of a sharing of a secret random value r^(k) where the sharing is guaranteed to be consistent (block 306), obtain a public random value R^(k) (that is consistent across all servers) (block 308), and compute a_(i) ^(k)=r_(i) ^(k)+R^(k)·x_(i) ^(k) (block 310). The purpose of R^(k) is to randomly modify the share provided by client C^(k) and thereby prevent two or more corrupted clients from colluding in a manner where they each individually submit invalid shares, but the invalid shares become valid when added together (which would break this protocol).

Upon completing the loop (block 312), server S_(i) can securely broadcast a_(i) ^(k) for k=1, . . . , N to the other servers (block 314) and receive a_(q) ^(k) for k=1, . . . , N from each other server S_(q) (block 316), thereby allowing server S_(i) to build sharings {right arrow over (a^(k))} for k=1, . . . , N.

At blocks 318 and 320, server S_(i) can initialize a binary tree T by ordering the a sharings at the leaves of T (such that {right arrow over (a^(k))} resides at the k-th leaf) and recursively storing {right arrow over (a_(L))}+{right arrow over (a_(R))} at an internal node where {right arrow over (a_(L))} and {right arrow over (a_(R))} are the left and right children of that internal node (resulting in a tree height of log N where the root is at level 0 and the leaves at level log N) (block 320). Server S_(i) can further mark the root of T and all nodes below the root as unresolved (block 322).

Server S_(i) can then initialize a set of corrupted servers {tilde over (S)}, a set of corrupted clients {tilde over (C)}, and a set of blocked clients Ĉ to null (0) (block 324) and flowchart 300 can proceed to FIG. 3B.

At block 326 of FIG. 3B, can enter a loop for each level l=0, . . . , log N of binary tree T. Within the level loop, server S_(i) can initialize a counter ctr_(q) for every server S_(q) (where q∈[n]) to zero (block 328), enter another loop for each node m at level l that is marked as unresolved (block 330), execute Reconstruct([a]) for the a sharing of node m (block 332), and check the outcome (block 334). If Reconstruct procedure succeeds and there are no bad points (i.e., the sharing is perfectly consistent), server S_(i) can mark the entire sub-tree rooted at node m as resolved (block 336).

If the Reconstruct procedure succeeds but there is a set of servers S′⊂S whose shares disagree with the reconstructed polynomial, server S_(i) can increment counter ctr_(q) for each server S_(q)∈′ (block 338). As part of this step, if there is a server S_(q) whose counter ctr_(q) has exceeded (1−ρ)N, server S_(i) can add server S_(q) to the list of corrupted servers {tilde over (S)} (which means that from this point, S_(q) can be safely ignored by all honest servers).

And if the Reconstruct procedure fails (i.e., no polynomial is found) and node m is the k-th leaf of T, server S_(i) can conclude that client C^(k) is corrupted and can add client C^(k) to the set of corrupted clients {tilde over (C)} (block 340). In this case, because C^(k) is considered “corrupted” rather than “blocked,” it is not entitled to the opportunity of using the stronger input scheme mentioned above.

Subsequently to blocks 336-340, server S_(i) can check whether all nodes at level l are now marked as resolved (block 342); if so, server S_(i) can exit the node loop. Otherwise, server S_(i) can reach the end of the current node loop iteration (block 344) and return to block 330 in order to process the next node m at level l. Further, upon completing the node loop, server S_(i) can reach the end of the current level loop iteration (block 346) and return to block 326 in order to process the next level l of binary tree T.

Upon completing the level loop, server S_(i) can, for each leaf of T comprising a sharing {right arrow over (a^(k))}that is still marked as unresolved, add client C^(k) (i.e., the client that provided the input x^(k) used to generate {right arrow over (a^(k))}) to the set of blocked clients Ĉ (block 348). Finally, at block 350, server S_(i) can output the set of blocked clients Ĉ, the set of corrupted clients {tilde over (C)}, and the set of corrupted servers {tilde over (S)} and flowchart 300 can end. As with flowchart 200 of FIG. 2 , the blocked clients in Ĉ can be given another opportunity to provide their respective shares via a stronger input scheme in which client-side cheating cannot be performed.

Certain embodiments described herein can employ various computer-implemented operations involving data stored in computer systems. For example, these operations can require physical manipulation of physical quantities—usually, though not necessarily, these quantities take the form of electrical or magnetic signals, where they (or representations of them) are capable of being stored, transferred, combined, compared, or otherwise manipulated. Such manipulations are often referred to in terms such as producing, identifying, determining, comparing, etc. Any operations described herein that form part of one or more embodiments can be useful machine operations.

Further, one or more embodiments can relate to a device or an apparatus for performing the foregoing operations. The apparatus can be specially constructed for specific required purposes, or it can be a generic computer system comprising one or more general purpose processors (e.g., Intel or AMD ×86 processors) selectively activated or configured by program code stored in the computer system. In particular, various generic computer systems may be used with computer programs written in accordance with the teachings herein, or it may be more convenient to construct a more specialized apparatus to perform the required operations. The various embodiments described herein can be practiced with other computer system configurations including handheld devices, microprocessor systems, microprocessor-based or programmable consumer electronics, minicomputers, mainframe computers, and the like.

Yet further, one or more embodiments can be implemented as one or more computer programs or as one or more computer program modules embodied in one or more non-transitory computer readable storage media. The term non-transitory computer readable storage medium refers to any data storage device that can store data which can thereafter be input to a computer system. The non-transitory computer readable media may be based on any existing or subsequently developed technology for embodying computer programs in a manner that enables them to be read by a computer system. Examples of non-transitory computer readable media include a hard drive, network attached storage (NAS), read-only memory, random-access memory, flash-based nonvolatile memory (e.g., a flash memory card or a solid state disk), a CD (Compact Disc) (e.g., CD-ROM, CD-R, CD-RW, etc.), a DVD (Digital Versatile Disc), a magnetic tape, and other optical and non-optical data storage devices. The non-transitory computer readable media can also be distributed over a network coupled computer system so that the computer readable code is stored and executed in a distributed fashion.

Finally, boundaries between various components, operations, and data stores are somewhat arbitrary, and particular operations are illustrated in the context of specific illustrative configurations. Other allocations of functionality are envisioned and may fall within the scope of the invention(s). In general, structures and functionality presented as separate components in exemplary configurations can be implemented as a combined structure or component. Similarly, structures and functionality presented as a single component can be implemented as separate components.

As used in the description herein and throughout the claims that follow, “a,” “an,” and “the” includes plural references unless the context clearly dictates otherwise. Also, as used in the description herein and throughout the claims that follow, the meaning of “in” includes “in” and “on” unless the context clearly dictates otherwise.

The above description illustrates various embodiments along with examples of how aspects of particular embodiments may be implemented. These examples and embodiments should not be deemed to be the only embodiments and are presented to illustrate the flexibility and advantages of particular embodiments as defined by the following claims. Other arrangements, embodiments, implementations and equivalents can be employed without departing from the scope hereof as defined by the claims. 

What is claimed is:
 1. A method comprising: receiving, by each server in a plurality of servers executing a secure multi-party computation (MPC) protocol, shares of a plurality of inputs to the MPC protocol from a plurality of clients, wherein each input in the plurality of inputs is private to each client in the plurality of clients, and wherein each share is generated from its corresponding input using a threshold secret sharing scheme; verifying, by said each server, whether the shares of the plurality of inputs are valid or invalid; determining, by said each server based on the verifying, a set of clients in the plurality of clients that may be corrupted; and sending, by said each server to each client in the set of clients, a request to resubmit the shares of the client's inputs using an alternative input scheme that is distinct from the threshold secret sharing scheme.
 2. The method of claim 1 further comprising: determining, for a share that is determined to be invalid, that a client that submitted the share is corrupted; and ignoring the input of the client during a computation phase of the MPC protocol.
 3. The method of claim 2 wherein determining that the client that submitted the share is corrupted comprises: determining that an input sharing submitted by the client is not consistent.
 4. The method of claim 1 further comprising: determining, for a share that is determined to be invalid, that a server that holds the share is corrupted; and preventing the server from participating in a computation phase of the MPC protocol.
 5. The method of claim 4 wherein determining that the server that holds the share is corrupted comprises: determining that the server holds more than a threshold number of invalid shares of consistent input sharings submitted by the plurality of clients.
 6. The method of claim 1 wherein there are n servers in the plurality of servers and at most t<n/4 servers are corrupted, and wherein there are N clients in the plurality of clients and at most (1−ρ)N clients are corrupted for 0<ρ≤1.
 7. The method of claim 1 wherein the shares are verified together for all clients in the plurality of clients.
 8. A non-transitory computer readable storage medium having stored thereon program code executable by each server in a plurality of servers executing a secure multi-party computation (MPC) protocol, the program code causing said each server to execute a method comprising: receiving shares of a plurality of inputs to the MPC protocol from a plurality of clients, wherein each input in the plurality of inputs is private to each client in the plurality of clients, and wherein each share is generated from its corresponding input using a threshold secret sharing scheme; verifying whether the shares of the plurality of inputs are valid or invalid; determining, based on the verifying, a set of clients in the plurality of clients that may be corrupted; and sending, to each client in the set of clients, a request to resubmit the shares of the client's inputs using an alternative input scheme that is distinct from the threshold secret sharing scheme.
 9. The non-transitory computer readable storage medium of claim 8 wherein the method further comprises: determining, for a share that is determined to be invalid, that a client that submitted the share is corrupted; and ignoring the input of the client during a computation phase of the MPC protocol.
 10. The non-transitory computer readable storage medium of claim 9 wherein determining that the client that submitted the share is corrupted comprises: determining that an input sharing submitted by the client is not consistent.
 11. The non-transitory computer readable storage medium of claim 8 wherein the method further comprises: determining, for a share that is determined to be invalid, that a server that holds the share is corrupted; and preventing the server from participating in a computation phase of the MPC protocol.
 12. The non-transitory computer readable storage medium of claim 11 wherein determining that the server that holds the share is corrupted comprises: determining that the server holds more than a threshold number of invalid shares of consistent input sharings submitted by the plurality of clients.
 13. The non-transitory computer readable storage medium of claim 8 wherein there are n servers in the plurality of servers and at most t<n/4 servers are corrupted, and wherein there are N clients in the plurality of clients and at most (1−ρ)N clients are corrupted for 0<ρ≤1.
 14. The non-transitory computer readable storage medium of claim 8 wherein the shares are verified together for all clients in the plurality of clients.
 15. A server among a plurality of servers executing a secure multi-party computation (MPC) protocol, the server comprising: a processor; and a non-transitory computer readable medium having stored thereon program code that, when executed, causes the processor to: receive shares of a plurality of inputs to the MPC protocol from a plurality of clients, wherein each input in the plurality of inputs is private to each client in the plurality of clients, and wherein each share is generated from its corresponding input using a threshold secret sharing scheme; verify whether the shares of the plurality of inputs are valid or invalid; determine, based on the verifying, a set of clients in the plurality of clients that may be corrupted; and send, to each client in the set of clients, a request to resubmit the shares of the client's inputs using an alternative input scheme that is distinct from the threshold secret sharing scheme.
 16. The server of claim 15 wherein the program code further causes the processor to: determine, for a share that is determined to be invalid, that a client that submitted the share is corrupted; and ignore the input of the client during a computation phase of the MPC protocol.
 17. The server of claim 16 wherein determining that the client that submitted the share is corrupted comprises: determining that an input sharing submitted by the client is not consistent.
 18. The server of claim 15 wherein the program code further causes the processor to: determine, for a share that is determined to be invalid, that a server that holds the share is corrupted; and prevent the server from participating in a computation phase of the MPC protocol.
 19. The server of claim 18 wherein determining that the server that holds the share is corrupted comprises: determining that the server holds more than a threshold number of invalid shares of consistent input sharings submitted by the plurality of clients.
 20. The server of claim 15 wherein there are n servers in the plurality of servers and at most t<n/4 servers are corrupted, and wherein there are N clients in the plurality of clients and at most (1−ρ)N clients are corrupted for 0<ρ≤1.
 21. The server of claim 15 wherein the shares are verified together for all clients in the plurality of clients. 